Problem: Find all the solutions to
\[\sqrt[3]{2 - x} + \sqrt{x - 1} = 1.\]Enter all the solutions, separated by commas.
Let $y = \sqrt[3]{2 - x}.$  Then $y^3 = 2 - x,$ so $x = 2 - y^3.$  Then
\[\sqrt{x - 1} = \sqrt{1 - y^3},\]so we can write the given equation as $y + \sqrt{1 - y^3} = 1.$  Then
\[\sqrt{1 - y^3} = 1 - y.\]Squaring both sides, we get $1 - y^3 = 1 - 2y + y^2,$ so $y^3 + y^2 - 2y = 0.$  This factors as $y(y - 1)(y + 2) = 0,$ so $y$ can be 0, 1, or $-2.$
These lead to $x = \boxed{1,2,10}.$  We check that these solutions work.